Question

Resistance of a metal wire of length $1$m is $26\Omega $ at $20^\circ$C. If the diameter of the wire is $0.3$ mm, what will be the resistivity of the metal at that temperature?

Answer 1

Hint: The resistivity of a substance is not dependent on its size or shape but its value is dependent on the nature of the substance and the temperature. Resistivity is used to determine how much a material opposes the flow of electric current. Low resistivity value indicates that the material is a good conductor of electricity. Using the proportionality of resistance to the length and area of a material, the value of resistivity can be determined.

Formula used: $R = \dfrac{{\rho L}}{A}$
\[
  R - {\text{ }}Resistance \\
  L - length \\
  A - {\text{ }}Area \\
  \rho - {\text{ }}Resistivity \]

Complete step-by-step solution:
Given:
Resistance of wire R = $26\Omega $
Length of wire L = $1$ mm
Diameter of the wire d= $0.3$mm
Temperature=$20^\circ$C
The given material is a wire that is cylindrical in shape. Therefore for area we can substitute A=$\pi {r^2}$.
To calculate the radius, we should use the equation $radius = \dfrac{{diameter}}{2}$
Therefore we get radius to be, $radius = \dfrac{{0.3}}{2} = 0.15mm$
In order to convert the value from millimeter to meter, we should multiply the value by ${10^{ - 3}}$
Hence $radius = 0.15mm = 0.15*{10^{ - 3}}m$
To find resistivity, $\rho $
Using the equation $R = \dfrac{{\rho L}}{A}$
Reordering the equation, $\rho = \dfrac{{RA}}{L}$
Substituting the values we get, $\rho=\dfrac{{R*\pi{r^2}}}{L}=\dfrac{{26*3.14*{{(0.15*{{10}^{ - 3}})}^2}}}{1}$
$\rho = 1.84*{10^{ - 8}}\Omega m$
The resistivity of the metal wire is $1.84*{10^{ - 8}}\Omega m$

Note: While solving questions of this type one should be careful to substitute the value of the radius and not the given value of the diameter. Radius$ = \dfrac{{diameter}}{2}$ equation should be used to modify it from diameter to radius. The units of the values must also be taken into consideration. All the values substituted in the equation should be in the same unit, preferably in the SI unit format. In this question, it is necessary to convert the unit of radius from millimeter to meter. The conversion factor \[1{\text{ m }} = 1000{\text{ mm}}\] should be used to convert the unit from millimeter to meter or vice-versa.